Open Access

Existence results for classes of infinite semipositone problems

  • Jerome GoddardII1,
  • Eun Kyoung Lee2,
  • Lakshmi Sankar3 and
  • R Shivaji4Email author
Boundary Value Problems20132013:97

DOI: 10.1186/1687-2770-2013-97

Received: 23 October 2012

Accepted: 5 April 2013

Published: 19 April 2013

Abstract

We consider the problem

{ Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equa_HTML.gif

where Δ p u = div ( | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq1_HTML.gif, p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, Ω is a smooth bounded domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq6_HTML.gif, γ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq7_HTML.gif and α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq8_HTML.gif. Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case c = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq9_HTML.gif is presented.

1 Introduction

Consider the nonsingular boundary value problem:
{ Δ u = a u b u 2 c h ( x ) , x Ω , u = 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ1_HTML.gif
(1)

where Ω is a smooth bounded domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq10_HTML.gif, Δ u = div ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq11_HTML.gif is the Laplacian of u and h : Ω ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq12_HTML.gif is a C 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq13_HTML.gif function satisfying h ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq14_HTML.gif for x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq15_HTML.gif, h ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq16_HTML.gif, max x Ω ¯ h ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq17_HTML.gif and h ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq18_HTML.gif for x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq19_HTML.gif. Existence of positive solutions of problem (1) was studied in [1]. In particular, it was proved that given an a > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq20_HTML.gif and b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif there exists a c ( a , b , Ω ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq21_HTML.gif such that for c < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq22_HTML.gif (1) has positive solutions. Here, λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq23_HTML.gif is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when a λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq24_HTML.gif. Later in [2], these results were extended to the case of the p-Laplacian operator, Δ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq25_HTML.gif, where Δ p u = div ( | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq1_HTML.gif, p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif. Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity f ( s , x ) = a s b s 2 c h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq26_HTML.gif satisfies f ( 0 , x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq27_HTML.gif for some x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq28_HTML.gif. See [39] for some existence results for semipositone problems.

In this paper, we study positive solutions to the singular boundary value problem:
{ Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ2_HTML.gif
(2)

where Δ p u = div ( | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq1_HTML.gif, p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, Ω is a smooth bounded domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq10_HTML.gif, α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, and γ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq30_HTML.gif. In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity f ( s ) = a s p 1 b s γ 1 c s α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq31_HTML.gif satisfies lim s 0 + f ( s ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq32_HTML.gif. One can refer to [1014], and [1517] for some recent existence results of infinite semipositone problems. We establish the following theorem.

Theorem 1.1 Given a , b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq33_HTML.gif, γ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq34_HTML.gif, and α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, there exists a constant c 1 = c 1 ( a , b , α , p , γ , Ω ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq35_HTML.gif such that for c < c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq36_HTML.gif, (2) has a positive solution.

Remark 1.1 In the nonsingular case ( α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq37_HTML.gif), positive solutions exist only when a > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq20_HTML.gif (the principal eigenvalue) (see [1, 2]). But in the singular case, we establish the existence of a positive solution for any given a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif.

Next, we study positive radial solutions to the problem:
{ Δ p u = K ( | x | ) ( a u p 1 b u γ 1 c u α ) , x Ω , u = 0 , if  | x | = r 0 , u 0 , as  | x | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ3_HTML.gif
(3)
where Ω = { x R n | | x | > r 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq38_HTML.gif is an exterior domain, n > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq39_HTML.gif, a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq40_HTML.gif, α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, γ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq30_HTML.gif and K : [ r 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq41_HTML.gif belongs to a class of continuous functions such that lim r K ( r ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq42_HTML.gif. By using the transformation: r = | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq43_HTML.gif and s = ( r r 0 ) n + p p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq44_HTML.gif, we reduce (3) to the following boundary value problem:
{ ( | u | p 2 u ) = h ( s ) ( a u p 1 b u γ 1 c u α ) , 0 < s < 1 , u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ4_HTML.gif
(4)

where h ( s ) = ( p 1 n p ) p r 0 p s p ( n 1 ) n p K ( r 0 s ( p 1 ) n p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq45_HTML.gif. We assume:

( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq46_HTML.gif) K C ( [ r 0 , ) , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq47_HTML.gif and satisfies K ( r ) < 1 r n + θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq48_HTML.gif for r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq49_HTML.gif, and for some θ such that ( n p p 1 ) α < θ < n p p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq50_HTML.gif.

With the condition ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq46_HTML.gif), h satisfies:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ5_HTML.gif
(5)

We note that if θ n p p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq51_HTML.gif then h ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq52_HTML.gif is nonsingular at 0 and h C ( [ 0 , 1 ] , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq53_HTML.gif. In this case, problem (4) can be studied using ideas in the proof of Theorem 1.1. Hence, our focus is on the case when θ < n p p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq54_HTML.gif in which, h may be singular at 0. Note that in this case h ˆ = inf s ( 0 , 1 ) h ( s ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq55_HTML.gif.

Remark 1.2 Note that ρ + α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq56_HTML.gif since θ > ( n p p 1 ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq57_HTML.gif.

We then establish the following theorem.

Theorem 1.2 Given a , b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq58_HTML.gif, γ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq34_HTML.gif, α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, and assume ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq46_HTML.gif) holds. Then there exists a constant c 2 = c 2 ( a , b , α , p , γ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq59_HTML.gif such that for c < c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq60_HTML.gif, (3) has a positive radial solution.

Finally, we prove a bifurcation result for the problem
{ Δ p u = a u p 1 b u γ 1 u α , x Ω , u = 0 , on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ6_HTML.gif
(6)

where Ω is a smooth bounded domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a is a positive parameter, b , α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq61_HTML.gif, p > 1 + α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq62_HTML.gif and γ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq30_HTML.gif. We prove the following.

Theorem 1.3 The boundary value problem (6) has a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq63_HTML.gif at ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq64_HTML.gif (as shown in Figure 1).
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig1_HTML.jpg
Figure 1

Bifurcation diagram, a vs. u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq65_HTML.gif for ( 6 ).

Our results are obtained via the method of sub-super solutions. By a subsolution of (2), we mean a function ψ W 1 , p ( Ω ) C ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq66_HTML.gif that satisfies
{ Ω | ψ | p 2 ψ w d x Ω a ψ p 1 b ψ γ 1 c ψ α w d x , for every  w W , ψ > 0 , in  Ω , ψ = 0 , on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equb_HTML.gif
and by a supersolution we mean a function Z W 1 , p ( Ω ) C ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq67_HTML.gif that satisfies:
{ Ω | Z | p 2 Z w d x Ω a Z p 1 b Z γ 1 c Z α w d x , for every  w W , Z > 0 , in  Ω , Z = 0 , on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equc_HTML.gif

where W = { ξ C 0 ( Ω ) | ξ 0  in  Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq68_HTML.gif. The following lemma was established in [13].

Lemma 1.4 (see [13, 18])

Let ψ be a subsolution of (2) and Z be a supersolution of (2) such that ψ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq69_HTML.gif in Ω. Then (2) has a solution u such that ψ u Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq70_HTML.gif in Ω.

Finding a positive subsolution, ψ, for such infinite semipositone problems is quite challenging since we need to construct ψ in such a way that lim x Ω Δ p ψ = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq71_HTML.gif and Δ p ψ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq72_HTML.gif in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form ψ = k ϕ 1 β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq73_HTML.gif, where k is an appropriate positive constant, β ( 1 , p p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq74_HTML.gif and ϕ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif is the eigenfunction corresponding to the first eigenvalue of Δ p ϕ = λ | ϕ | p 2 ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq76_HTML.gif in Ω, ϕ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq77_HTML.gif on Ω.

In Sections 2, 3, and 4, we provide proofs of our results. Section 5 is concerned with providing some exact bifurcation diagrams of positive solutions of (2) when Ω = ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq78_HTML.gif and p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq79_HTML.gif.

2 Proof of Theorem 1.1

We first construct a subsolution. Consider the eigenvalue problem Δ p ϕ = λ | ϕ | p 2 ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq80_HTML.gif in Ω, ϕ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq77_HTML.gif on Ω. Let ϕ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif be an eigenfunction corresponding to the first eigenvalue λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq23_HTML.gif such that ϕ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq81_HTML.gif and ϕ 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq82_HTML.gif. Also, let δ , m , μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq83_HTML.gif be such that | ϕ 1 | m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq84_HTML.gif in Ω δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq85_HTML.gif and ϕ 1 μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq86_HTML.gif in Ω Ω δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq87_HTML.gif, where Ω δ = { x Ω | d ( x , Ω ) δ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq88_HTML.gif. Let β ( 1 , p p 1 + α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq89_HTML.gif be fixed. Here, note that since α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, p p 1 + α > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq90_HTML.gif. Choose a k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq91_HTML.gif such that 2 b k γ p + β p 1 λ 1 k α a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq92_HTML.gif. Define c 1 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq93_HTML.gif. Note that c 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq94_HTML.gif by the choice of k and β. Let ψ = k ϕ 1 β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq95_HTML.gif. Then
Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equd_HTML.gif
To prove ψ is a subsolution, we need to establish:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ7_HTML.gif
(7)
in Ω if c < c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq36_HTML.gif. To achieve this, we split the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq96_HTML.gif into three, namely,
k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Eque_HTML.gif
Now to prove (7) holds in Ω, it is enough to show the following three inequalities:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ8_HTML.gif
(8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ10_HTML.gif
(10)
From the choice of k, ( a β p 1 λ 1 k α ) 2 b k γ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq97_HTML.gif, hence,
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ11_HTML.gif
(11)
Using ϕ 1 μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq98_HTML.gif in Ω Ω δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq87_HTML.gif and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq99_HTML.gif
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 ) 2 k α ϕ 1 α β c k α ϕ 1 α β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ12_HTML.gif
(12)
Finally, since | ϕ 1 | m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq100_HTML.gif, in Ω δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq85_HTML.gif, and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq101_HTML.gif,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 p β ( p 1 ) α β c k α ϕ 1 α β ϕ 1 p β ( p 1 + α ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equf_HTML.gif
Since p β ( p 1 + α ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq102_HTML.gif,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c k α ϕ 1 α β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ13_HTML.gif
(13)

From (11), (12) and (13) we see that equation (7) holds in Ω, if c < c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq36_HTML.gif. Next, we construct a supersolution. Let e be the solution of Δ p e = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq103_HTML.gif in Ω , e = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq104_HTML.gif on Ω. Choose M ¯ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq105_HTML.gif such that a u p 1 b u γ 1 c u α M ¯ p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq106_HTML.gif u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq107_HTML.gif and M ¯ e ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq108_HTML.gif. Define Z = M ¯ e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq109_HTML.gif. Then Z is a supersolution of (2). Thus, Theorem 1.1 is proven.

3 Proof of Theorem 1.2

We begin the proof by constructing a subsolution. Consider
( | ϕ | p 2 ϕ ) = λ | ϕ | p 2 ϕ , t ( 0 , 1 ) , ϕ ( 0 ) = ϕ ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ14_HTML.gif
(14)
Let ϕ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif be an eigenfunction corresponding to the first eigenvalue of (14) such that ϕ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq81_HTML.gif and ϕ 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq110_HTML.gif. Then there exist d 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq111_HTML.gif such that 0 < ϕ 1 ( t ) d 1 t ( 1 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq112_HTML.gif for t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq113_HTML.gif. Also, let ϵ < ϵ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq114_HTML.gif and m , μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq115_HTML.gif be such that | ϕ 1 | m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq116_HTML.gif in ( 0 , ϵ ] [ 1 ϵ , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq117_HTML.gif and ϕ 1 μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq86_HTML.gif in ( ϵ , 1 ϵ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq118_HTML.gif. Let β ( 1 , p ρ p 1 + α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq119_HTML.gif be fixed and choose k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq91_HTML.gif such that 2 b k γ p + β p 1 λ 1 k α h ˆ a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq120_HTML.gif. Define c 2 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq121_HTML.gif. Then c 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq122_HTML.gif by the choice of k and β. Let ψ = k ϕ 1 β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq73_HTML.gif. This implies that:
( | ψ | p 2 ψ ) = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equg_HTML.gif
To prove ψ is a subsolution, we need to establish:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ15_HTML.gif
(15)
Here, we note that the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = h ˆ k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) h ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq123_HTML.gif h ( t ) ( a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq124_HTML.gif, where h ˆ = inf s ( 0 , 1 ) h ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq125_HTML.gif. Now to prove (15) holds in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq126_HTML.gif, it is enough to show the following three inequalities:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ16_HTML.gif
(16)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ17_HTML.gif
(17)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ18_HTML.gif
(18)
From the choice of k, ( a β p 1 λ 1 k α h ˆ ) 2 b k γ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq127_HTML.gif, hence,
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ19_HTML.gif
(19)
Using ϕ 1 μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq98_HTML.gif in ( ϵ , 1 ϵ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq118_HTML.gif and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq128_HTML.gif
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 h ˆ ) 2 k α ϕ 1 α β c k α ϕ 1 α β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ20_HTML.gif
(20)
Next, we prove (18) holds in ( 0 , ϵ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq129_HTML.gif. Since | ϕ 1 | m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq130_HTML.gif, and p β ( p 1 ) > α β + ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq131_HTML.gif
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 ρ k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β d 1 ρ t ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equh_HTML.gif
Since h ( t ) 1 t ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq132_HTML.gif in ( 0 , ϵ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq129_HTML.gif, and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq133_HTML.gif,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c h ( t ) k α ϕ 1 α β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ21_HTML.gif
(21)

Proving (18) holds in [ 1 ϵ , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq134_HTML.gif is straightforward since h is not singular at t = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq135_HTML.gif. Thus, from equations (19), (20) and (21), we see that (15) holds in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq126_HTML.gif. Hence, ψ is a subsolution. Let Z = M ¯ e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq109_HTML.gif where e satisfies ( | e | p 2 e ) = h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq136_HTML.gif in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq137_HTML.gif, e ( 0 ) = e ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq138_HTML.gif and M ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq139_HTML.gif is such that a u p 1 b u γ 1 c u α M ¯ p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq140_HTML.gif u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq107_HTML.gif and M ¯ e ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq108_HTML.gif. Then Z is a supersolution of (4) and there exists a solution u of (4) such that u [ ψ , Z ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq141_HTML.gif. Thus, Theorem 1.2 is proven.

4 Proof of Theorem 1.3

We first prove (6) has a positive solution for every a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif. We begin by constructing a subsolution. Let ϕ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif be as in the proof of Theorem 1.1 (see Section 2). Let β ( 1 , p p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq74_HTML.gif, and choose a k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq91_HTML.gif such that b k γ p + β p 1 λ 1 k α a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq142_HTML.gif. Let ψ = k ϕ 1 β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq73_HTML.gif. Then
Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equi_HTML.gif
To prove ψ is a subsolution, we will establish:
k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) a k p 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ22_HTML.gif
(22)
in Ω. To achieve this, we rewrite the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq96_HTML.gif as k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq143_HTML.gif. Now to prove (22) holds in Ω, it is enough to show k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( γ 1 α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq144_HTML.gif. From the choice of k, ( a β p 1 λ 1 k α ) b k γ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq145_HTML.gif, hence,
k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equj_HTML.gif

Thus, ψ is a subsolution. It is easy to see that Z = ( a b ) 1 γ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq146_HTML.gif is a supersolution of (6). Since k, can be chosen small enough, ψ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq69_HTML.gif. Thus, (6) has a positive solution for every a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif. Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq147_HTML.gif is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq63_HTML.gif at ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq64_HTML.gif.

5 Numerical results

Consider the boundary value problem
{ u ( x ) = a u b u 2 c u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ23_HTML.gif
(23)
where a , b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq58_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq40_HTML.gif and α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif. Using the quadrature method (see [19]), the bifurcation diagram of positive solutions of (23) is given by
G ( ρ , c ) = 0 ρ d s [ 2 ( F ( ρ ) F ( s ) ) ] = 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ24_HTML.gif
(24)
where F ( s ) : = 0 s f ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq148_HTML.gif where f ( t ) = a t b t 2 c t α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq149_HTML.gif and ρ = u ( 1 2 ) = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq150_HTML.gif. We plot the exact bifurcation diagram of positive solutions of (23) using Mathematica. Figure 2 shows bifurcation diagrams of positive solutions of (23) when a = 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq151_HTML.gif ( < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq152_HTML.gif) and b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq153_HTML.gif for different values of α.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig2_HTML.jpg
Figure 2

Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq154_HTML.gif , b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq155_HTML.gif .

Bifurcation diagrams of positive solutions of (23) when a = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq156_HTML.gif ( > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq157_HTML.gif) and b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq153_HTML.gif for different values of α is shown in Figure 3.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig3_HTML.jpg
Figure 3

Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq158_HTML.gif , b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq155_HTML.gif .

Finally, we provide the exact bifurcation diagram for (6) when p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq79_HTML.gif, and Ω = ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq78_HTML.gif. Consider
{ u ( x ) = a u b u 2 u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ25_HTML.gif
(25)
where a , b , α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq159_HTML.gif. The bifurcation diagram of positive solutions of (25) is given by
G ˜ ( ρ , a ) = 0 ρ d s [ 2 ( F ˜ ( ρ ) F ˜ ( s ) ) ] = 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ26_HTML.gif
(26)
where F ˜ ( s ) : = 0 s f ˜ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq160_HTML.gif where f ˜ ( t ) = a t b t 2 t α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq161_HTML.gif and ρ = u ( 1 2 ) = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq150_HTML.gif. The bifurcation diagram of positive solutions of (25) as well as the trivial solution branch are shown in Figure 4 when α = 0.5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq162_HTML.gif and b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq153_HTML.gif.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig4_HTML.jpg
Figure 4

Bifurcation diagram, a vs. ρ for ( 25 ) with α = 0.5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq163_HTML.gif , b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq155_HTML.gif .

Declarations

Acknowledgements

EK Lee was supported by 2-year Research Grant of Pusan National University.

Authors’ Affiliations

(1)
Department of Mathematics, Auburn University Montgomery
(2)
Department of Mathematics Education, Pusan National University
(3)
Department of Mathematics & Statistics, Mississippi State University
(4)
Department of Mathematics & Statistics, University of North Carolina at Greensboro

References

  1. Oruganti S, Shi J, Shivaji R: Diffusive logistic equation with constant harvesting, I: steady states. Trans. Am. Math. Soc. 2002, 354(9):3601-3619. 10.1090/S0002-9947-02-03005-2MATHMathSciNetView Article
  2. Oruganti S, Shi J, Shivaji R: Logistic equation with the p -Laplacian and constant yield harvesting. Abstr. Appl. Anal. 2004, 9: 723-727.MathSciNetView Article
  3. Ambrosetti A, Arcoya D, Biffoni B: Positive solutions for some semipositone problems via bifurcation theory. Differ. Integral Equ. 1994, 7: 655-663.MATH
  4. Anuradha V, Hai DD, Shivaji R: Existence results for superlinear semipositone boundary value problems. Proc. Am. Math. Soc. 1996, 124(3):757-763. 10.1090/S0002-9939-96-03256-XMATHMathSciNetView Article
  5. Arcoya D, Zertiti A: Existence and nonexistence of radially symmetric nonnegative solutions for a class of semipositone problems in an annulus. Rend. Mat. Appl. 1994, 14: 625-646.MATHMathSciNet
  6. Castro A, Garner JB, Shivaji R: Existence results for classes of sublinear semipositone problems. Results Math. 1993, 23: 214-220. 10.1007/BF03322297MATHMathSciNetView Article
  7. Castro A, Shivaji R: Nonnegative solutions for a class of nonpositone problems. Proc. R. Soc. Edinb. 1998, 108(A):291-302.MathSciNet
  8. Castro A, Shivaji R: Nonnegative solutions for a class of radially symmetric nonpositone problems. Proc. Am. Math. Soc. 1989, 106(3):735-740.MATHMathSciNet
  9. Castro A, Shivaji R: Positive solutions for a concave semipositone Dirichlet problem. Nonlinear Anal. 1998, 31: 91-98. 10.1016/S0362-546X(96)00189-7MATHMathSciNetView Article
  10. Ghergu M, Radulescu V: Sublinear singular elliptic problems with two parameters. J. Differ. Equ. 2003, 195: 520-536. 10.1016/S0022-0396(03)00105-0MATHMathSciNetView Article
  11. Hai DD, Sankar L, Shivaji R: Infinite semipositone problems with asymptotically linear growth forcing terms. Differ. Integral Equ. 2012, 25(11-12):1175-1188.MATHMathSciNet
  12. Hernandez J, Mancebo FJ, Vega JM: Positive solutions for singular nonlinear elliptic equations. Proc. R. Soc. Edinb. 2007, 137A: 41-62.MathSciNet
  13. Lee E, Shivaji R, Ye J: Classes of infinite semipositone systems. Proc. R. Soc. Edinb. 2009, 139(A):853-865.MATHMathSciNetView Article
  14. Lee E, Shivaji R, Ye J: Positive solutions for elliptic equations involving nonlinearities with falling zeros. Appl. Math. Lett. 2009, 22: 846-851. 10.1016/j.aml.2008.08.020MATHMathSciNetView Article
  15. Ramaswamy M, Shivaji R, Ye J: Positive solutions for a class of infinite semipositone problems. Differ. Integral Equ. 2007, 20(11):1423-1433.MATHMathSciNet
  16. Shi J, Yao M: On a singular nonlinear semilinear elliptic problem. Proc. R. Soc. Edinb. 1998, 128A: 1389-1401.MathSciNetView Article
  17. Zhang Z: On a Dirichlet problem with a singular nonlinearity. J. Math. Anal. Appl. 1995, 194: 103-113. 10.1006/jmaa.1995.1288MATHMathSciNetView Article
  18. Cui S: Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems. Nonlinear Anal. 2000, 41: 149-176. 10.1016/S0362-546X(98)00271-5MATHMathSciNetView Article
  19. Laetsch T: The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. Math. J. 1970, 20: 1-13. 10.1512/iumj.1970.20.20001MATHMathSciNetView Article

Copyright

© Goddard II et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.